SEMI-STATIC OBJECT DETECTION USING POLYGONAL MAPS FOR SAFE NAVIGATION OF INDUSTRIAL ROBOTS

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1 SEMI-STATIC OBJECT DETECTION USING POLYGONAL MAPS FOR SAFE NAVIGATION OF INDUSTRIAL ROBOTS Dario Lodi Rizzini, Gionata Boccalini, and Stefano Caselli Dipartiento di Ingegneria dell Inforazione, University of Para, viale G.P. Usberti 8A, 44 Para, Italy {dlr, gbocca, caselli Keywords: Abstract: Mapping, Range Sensing The collision and safety control of industrial UGVs equipped with laser range finders is often based on conservative area-oriented policies that lack in flexibility and does not deal well with non epheeral environent changes due to sei-static objects (e.g. passive isplaced objects). In this paper, we propose a ethod to detect and represent sei-static objects using polygonal local aps in order to iprove robot navigation. Each local ap consists of polylines representing the boundary of an object detected inside a safety area. Polylines are extracted fro laser scans and associated with the polylines of a reference ap using a siilarity easure criterion. Finally, the ap is updated by erging the new polylines. The proposed polygonal representation allows the recognition of new sei-static obstacles in the environent and supports ore flexible policies for safe navigation. An EKF localizer using artificial landarks and a fixed path navigation syste have been ipleented to replicate the navigation syste of industrial UGVs. The precision of environent reconstruction has been assessed with experients in siulated and real environents. INTRODUCTION Unanned Ground Vehicles (UGVs) are ore and ore used to anage logistics, individually or in coordinated teas, in industrial plants and warehouses. Being copletely autonoous, these vehicles are usually equipped with laser range finders to perceive obstacles and to localize and navigate in the environent. Laser range finders for localization are usually ounted on the top of LGV to avoid occlusions and to perceive artificial landarks with high reission value that have been previously placed in the environent. An Extented Kalan Filter (EKF) localizer tracks and estiates the pose of the robot using the observed landarks and an a priori ap of the landarks. Safety lasers (e.g. Sick S) are usually placed slightly above floor level for the detection of any object lying in the proxiity of the robot. One coon safety policy consists of partitioning the space in front of the laser idle, warning and protective areas. The warning and the protective areas are supposed to be free in noral working conditions so an appropriate size for this area ust be defined for each segent of the LGV trajectory. When an object is perceived in the warning area (shown in yellow in botto of Figure ), the UGV slows down according to the warning. When the protective area (in red in Figure ) is not free, the vehicle stops. For any plant operating with LGVs, the diension of warning and protective areas vary in the different regions of the working environent. This strict and conservative policy grants the safety of people and things, but it is inefficient and inflexible. Sizing the safety areas is a tie-consuing and tedious operation that ust be repeated whenever the environent layout changes. Furtherore, while it is andatory that the robot stops when a collision with a person or an obstacle ay occur, the efficiency of the syste is often affected by objects thoughtlessly left in the assued free space. Although the area classification schee has been conceived to ensure safety argin in presence of huan operators, the ajority of reduction in UGV navigation speed is actually due to passive isplaced objects. These objects ay e odeled as sei-static objects that do not belong to the persistent structure of the environent. The sei-static objects change their location with low frequency (e.g. after several hours) and while the robot is not observing the. In this paper, we propose a ethod to detect and represent sei-static objects using polygonal local aps in order to iprove robot navigation. In the proposed solution, the representation of sei-static objects is assigned to a collection of local aps where

2 each local ap covers a specific path segent. Local aps are ore anageable than a single global ap and can be ore easily integrated with the existent navigation syste. Such aps consist of polygonal lines or, shortly, polylines that represent the boundary of the obstacles found inside the warning and protective areas. Polygonal representation is eory and bandwidth inexpensive, copact and seantically rich. A siilarity easure is used to associate the observed polylines with a preexisting ap for robust detection of sei-static objects. When a sei-static object is found inside a safety area, the corresponding local ap is updated and the diension of the safety areas of the laser range sensor ay be adapted to the reduced free space. Thus, the next LGV visiting the sae region is not forced to slow down or to stop since the updated area does not contain sei-static object. More advanced and efficient policies ay be ipleented by giving up the conservative safety area. The proposed approach has been assessed in siulation and in a real experiental setup. For this purpose, an EKF localizer based on landark and a navigation syste have been ipleented. RELATED WORK Research on robot localization and ap building early et the proble of representing dynaics in the environent. The first and ost coon approach is the reoval of oving objects when detected. Fox et al. (Fox et al., 999) proposed a sensor odel for range finders that can deal with dynaic objects by odeling the as noise. The work of Wolf et al. (Wolf and Sukhate, 5) is one of the first siultaneous localization and apping (SLAM) ethods that explicitly distinguishes between dynaic entities and static environents using two different occupancy grid aps. Stachniss et al. (Stachniss and Burgard, 5) focus on the sei-static objects, i.e. objects that infrequently change their location while the robot is not observing the. Sei-static objects like doors, parking locations, etc., often define the discrete state of the environent providing also longter seantic inforation. Recent localization ethods represent static, sei-static and dynaic objects in the ap (Meyer-Delius et al., ). The ajority of these techniques are based on an occupancy grid ap, which is suitable to integrate range easureents into a robust representation of the environent, but it does not carry shape or seantic inforation (an object corresponds to a group of cells) and is not apt to be shared aong distributed robots. Several other works identifies either generic (Katz et al., 8) or specific oving entities like people (Spinello et al., 9) fro consecutive range easureents. Tipaldi and Raos (Tipaldi and Raos, 9) proposed a ethod perforing siultaneously scan atching and oving cluster detection using Conditional Rando Fields (CRF). These approaches are oriented only to the detection and shortter tracking of people and objects and represent their targets as clusters of points. Strict-sense polygonal representations have not been extensively used for apping, since a polyline is a coposite eleent and cannot be represented with a fixed nuber of variables as would be convenient for state estiation algoriths like Bayesian filters. On the other hand, EKF localization and apping algoriths often utilize line segent feature aps according to odels like SP-ap odels (Arras, ). Latecky et al. (Latecki et al., 4) illustrated a techniques for integrating the polylines extracted by range easureents by exploiting shape siilarity easure and scan order. POLYGONAL MAP REPRESENTATION This work ais to detect and track the non epheeral changes occurring inside the region of environent covered by the robot range sensors. The robot oves along a collection of path segents defined a priori. On each path segent a specific safety area (consisting of warning and protective areas) is defined to detect potential obstacles without including the static part of environent. When a stationary entity is detected inside a safety area, the syste should record such sei-static object in a ap of the environent. Instead of anaging a single global ap, the natural segentation of the robot path can be exploited to cover the working environent with ore anageable local aps of liited diensions. The reference fraes of these local aps are anchored to the reference frae on the path segents. A local ap consists of polygonal lines or polylines that are represented by the list of their vertices. Polylines are copact geoetric features that represent the countour of obstacles and that can be easily extracted by laser range scans. Polygonal aps eet several of the syste requireents. First, polylines require liited storage and bandwidth for their transission aong the LGVs working together in the environent. Second, the result of polylines extraction fro laser scan is already a segented set since each polyline represents an object or a portion of an ob-

3 P P Laser scan Polyline extraction Polyline association Polyline erge Polyline join Map update P.5.5 Point of view P New polyline points Point of view P New polyline points New polyline Warning area Protective area Map Current scan Robot poses P P Point of view P P New polyline points Point of view P P New polyline points New polyline Figure : Diagra illustrating the algorith for building polygonal aps fro laser scans. ject, though the extraction algorith ust handle errors due to occlusions and sensor noise. Finally, polylines encode shape inforation that would be iplicit in occupancy grid representations and that can be exploited for robust data association. The steps of the algorith for extracting and integrating the polylines fro the laser scan into a consistent ap are illustrated in Figure. The algorith extracts polylines fro current laser scan (polyline extraction) and associates the with the polylines contained in the existing ap using the siilarity easure (polyline association). Then, the scan polylines are erged with the associated eleents of the ap (polyline erge). Finally, the close polygonal lines are joined to reove inconsistent and redundant ites (polyline join). In the following, the details of the proposed algorith are discussed. P4 P4 P P 4 P P P. Polyline Extraction Polylines are extracted fro each raw laser range scan by reoving outliers, by splitting the scan into intervals and finally by choosing the vertices of the polylines. Scans are coonly represented as sorted vectors of ranges or, equivalently, of points (sybols r i and p i will refer to range or Cartesian point hence after). The iplicit angular order of ranges is used by the standard extraction techniques (Xavier et al., 5) as those applied in this work. First processing step is the statistical outlier reoval. For each window of ranges r i k,...,r i,...,r i+k centered on i th range the ean value µ i and standard deviation σ i are coputed. The two statistics are used to detect isolated easureents. If the difference between range r i and window ean value µ i is greater than a threshold α σ i (α is a paraeters set by the user), then easureent i is reoved fro the scan. The scan is then split into intervals in correspondence to a strong discontinuity between consecutive ranges. The discontinuity is detected according to the standard adaptive threshold criterion. The line connecting two consecutive discontinuous points represents the border of the occluded region. Each scan interval corresponds to a polyline representing an object or a part of an object. All the points of an interval ay serve as vertices of the polyline. However, when points are approxiatively aligned, the polyline is better represented by a less redundant list of points. Such result can be easily achieved using Douglas-Peuker algorith for curve siplification (Douglas and Peuker, 97). A siilar result could be obtained using the discrete curve evolution (DCE) proposed in (Latecki et al., 4).. Metric for Polylines The association of polylines belonging to different sets is a crucial operation for the construction of local ap and for the detection of the objects that have been already observed or not yet observed. Association is perfored on two sets of polylines, i.e. the polylines extracted fro the laser scan and the polylines belonging to the existing local ap. Polyline association requires a etric to easure the likeness of a single pair of polylines and a procedure to estiate the ost consistent joint set of associated pairs. The latter aspect is discussed in the next section. In this work, two different etrics are used. The first one ais at easuring the shape siilarity between two polylines. The second estiates a bound on the Euclidean distance between the points of two

4 Hystogra Polyline Hystogra Polyline Figure : Polyline shape siilarity of polyline (top-left) and polyline (botto-left) is easured coparing their respective tangent space histogras (top- and botto-right). polylines to validate the siilarity. Since the robot pose can be accurately estiated using a localizer, the pose of polylines in the ap is accurate enough to use proxiity as a validation of data association. Two siple curves in the plane are perceived as siilar if their tangent vector turns in the sae way. The concept of siilarity has been originally foralized in (Latecki and Lakaeper, 999) for closed polygons and then extended to polylines (Latecki et al., 4). The idea is to copare the noralized tangent histogras of two polylines P and P. The polygonal curves P s and P s are paraeterized w.r.t. their noralized arclength s [, ]. The tangent function T P (s) : [,] [,π] (and T P (s)) returns the angle between the axis x and the tangent vector to the polyline in P (s) (and P (s)). Since the polyline consists of edges and the tangent angle on each edge is constant, the tangent space function has the appearance of a histogra. Figure illustrates the noralized tangent space histogras of two different polylines. Tangent histogra is invariant to translation and a rotation applied to a polyline shifts up or down all the values of the corresponding tangent histogra, but the difference between tangent angles reains invariant. Furtherore, tangent histogra is not affected by the scale of the polylines since it is paraeterized w.r.t. the noralized arc length. Thus, tangent histogras can be used to define a etric that is invariant to both rigid otion and scale. Siilarity easure S(P,P ) of polylines P and P is forally defined by equation [ s(p,p ) = (T P (s) T P (s) + θ ) l ds ax, l ] l l () θ = (T P (s) T P (s))ds () where l and l are the lengths respectively of the polygons P and P and θ is the angle that iniizes the ean difference between the histogras. The second ter of equation () is introduced to weight the length of the polylines in the siilarity that would be otherwise invariant to scale. The above integral is concretely coputed on a refineent of the arclength intervals Map Observation Figure : An exaple of the coputation of restricted polylines P P and P P : the red arrows point out the vertices of P not belonging to P P. The invariance of siilarity easure is clearly an advantage beacuse repeated observations of the sae object are seldo gathered fro the sae point of view. However, if the object shape is abigous or not enough distinctive, siilarity easure ay be isleading. The association procedure defined in the following ay attenuate such proble, but a validation of siilarity that takes into account the distance between polylines should be perfored through a proper etric. Hausdorff distance provides a worst case estiate of the distance between two polylines P and P and is often used to copute the distance between two coposite geoetric sets. When two polylines represent the sae object and one of the is obtained fro a partial view of the object, the Hausdorff distance ay overestiate the Euclidean displaceent between the polylines. To overcoe this inconvenience we define a etric that copares the distance of two polylines restricted to the overlapping part. The definition is based on the correspondence between the index order of vertices and the angular order of the original laser scan points that reain in the polylines also after several erges. Let P and P be two polylines respectively with vertices p,, p,,..., p,n and p,, p,,..., p,n. Let i k be the index of the segent of P closer to vertex p,k of polygon P or, forally, i k argin d(p,k ; p, j p, j ) () j=,...,n The iniu and axiu indices are defined as follows i in = in{ i,..., i n } and i ax = ax{ i,..., i n }. Thus, the polyline P P restricted to P consists of the vertices p i with i = i in,..., i ax. Siilarly, we define the polyline P P restricted to P. Figure illustrates the effect of restriction on polylines. Thus, the restricted Hausdorff

5 distance is defined as the Hausdorff distance of the restricted polylines and can be used to liit the risk of isleading estiations.. Polyline Association The etrics defined in the previous section can be used to associate the polylines extracted fro the scan with the polylines contained in the existing local ap. The natural order of the polylines in the laser scan can be exploited to reduce the risk of wrong association. Map polylines can also be dynaically sorted according to the current viewpoint of the robot. Such constrained order also reduces the coplexity of data association proble and that be solved using dynaic prograing optiization as suggested in (Latecki et al., 4). Let S i = {S,S,...,S i } be the sorted list of the scan polylines fro the first to the i th (i =,...,n) and M j = {M,M,...,M j } be the sorted list of the ap polylines till the j th ( j =,...,). The solution of polyline association proble restricted to S i and M j is given by the sorted list of index pairs A i j and the total distance between the associated polylines is d(a i j ) = s(sī,m j ) (4) (ī, j) A i j where s( ) is the siilarity easure of polylines discussed in previous section. The solution A i j is the list of pairs that iniizes the distance of equation (4) of the restricted proble. Given two distinct association pairs (i, j ),(i, j ) A i j the strict condition i i and j j, briefly (i, j ) < (i, j ), ust hold (without loosing generality, otherwise swap i with i and j with j ). Hence, the best association A i j can be estiated considering the solution of sub-probles A i, j, A i, j and A i, j. In particular, the total distance of the solution is such that A i j = {(i, j)} argin d(a) (5) A i, j ;A i, j ;A i, j Observe that according to the above definition A i j contains any-to-any associations. The procedure is iterated until the final association set A n is achieved. Since at each iteration a new pairs is inserted according to equation (5), the final set contains pairs that have been forcibly included in spite of their siilarity. Thus, the restricted Hausdorff distance can be used to validate the associations contained in A n by fixing a threshold on the axiu distance..4 Merging Polylines in the Map Once the polylines fro the scan are associated to the the polylines in the ap, the ap is updated by erging the associated polygonal ites and by inserting the unassociated ones. The counterclockwise scan order around the viewpoint discussed previously is also exploited to erge the polylines. The polylines directly extracted fro a scan are already sorted and the ap polylines are sorted counterclockwise w.r.t. the current viewpoint. Our algorith for polyline erging scans the two sorted lists, while keeping the pointer P to the current ap polyline aligned with the current scan polyline P. When P and P overlap, the vertices of polyline P are projected on P (i.e. on the ap) and the resulting erg polyline is soothed using Douglas-Peucker algorith. Otherwise, P and P are respectively kept or added to the ap. The previously illustrated union of polylines produces a new collection of polylines fro the old local ap and the features extracted fro the scan. The new ap ay contains overlapping polylines or sall inconsistencies caused by partial views of the sae object and by inaccuracies in polygon extraction. To correct such faults and to reduce the size of the ap, such polylines are joined together. First, the polylines are sorted once again according to the sae scan viewpoint order to detect overlapping parts through a linear scan of the polylines list. The overlapping ites are found using the restricted Hausdorff distance. Second, the joining ethod is siilar to the erging algorith is applied. 4 SEMI-STATIC MAPS The algorith described in the previous section can be used to build a polygonal ap fro a sequence of laser scans. For each path segent, the robot keeps a local ap consisting of polylines that contains the objects found inside the safety areas. Since the ai of this work is to iprove area based policies for industrial environents, only the polylines that intersect the safety areas are considered. Furtherore, the proposed technique takes care of erging and joining overlapping polylines. Thus, we can assue that each polyline represents a distinguishable sei-static object of the environent. Every tie an UGV visits a region of the environent, polylines are extracted fro each scan and associated with the polylines of the local ap corresponding to this region. The data association technique described in section. is crucial for the detection of environental changes. In particular, the siilarity easure and the scan order help to identify the candidate association pairs and the reduced Hausdorff distance is used for the validation according to a validation gate criterion. The results presented in section 6 show that a convenient acceptance threshold for our experiental setup is slightly above c. The unassociated scan polylines are classified as new

6 sei-static objects and inserted into the local ap. The unassociated polylines in the ap are arked as blank and reoved after they have not been not observed for k ties. The siple sei-static ap can be used to define new navigation policies. In particular, when seistatic obstacles are detected, the syste can reduce one or both the safety areas (the warning and protective areas) for the path segents so that no obstacle lies inside these aps. A less conservative and ore flexible rule could copare the polylines without regard to the areas and classify the obstacle as an already observed obstacle that can be ignored. However, the ai of this work is not to define a policy, but to provide the eans for efficiently dealing with non static environents to the developers of the navigation syste. 5 LOCALIZATION AND NAVIGATION 5. EKF Localization The standard localization syste of industrial LGVs works with artificial landarks that can easily detected by laser range scanners due to the reission value of landark aterial. Localization is fundaental to perfor robot navigation and also to build the local aps as illustrated in section. Navigation based on path following requires an accurate easure of pose displaceent between the current and the desired pose to perfor the proper correction as will be discussed in the next subsection. A correct estiate of robot pose allows consistent positioning of the polylines in the local ap. This work ais at assessing the viability and effectiveness of sei-static object detection in the working condition of industrial obile robots. For this purpose, a localization syste based on artificial landarks has been setup in our laboratory. Artificial landarks have been placed in the environent as shown in Figure 4(a)-(c) and a ap containing the position of landarks is available. The details of the experiental setup will be described in section 6. Given the approxiate inforation about the group of arkers visibile at the robot location, a triangulation algorith (Esteves et al., ) provides the initial estiate of the robot pose. The algorith requires three visible landarks to provide the estiate of pose since it belongs to the category of three object triangulation ethods. The triangulation also returns the estiation of covariance atrix representing the uncertainty on pose. The Extended Kalan Filter (EKF) is the core coponent of the localizer and has the responsibility of keeping updated the estiate of the robot pose. The EKF uses a prediction odel based on reference frae transforation rather than a velocity odel and a point-landark sensor odel. In our setup, an iteration of EKF including prediction and correction is perfored in about 4 s and is rather efficient. The state of the localizer representing the robot pose is referred to a unique global reference syste instead of the local ap frae. Currently, the localizer is copletely independent fro the ap building coponent. Such solution is convenient since it does not interfere with the preexistent localization syste. In future works, the polygonal features of the local ap could be used by the EKF within the artificial arkers, e.g. adopting an extended SP-ap odel (Arras, ). 5. Navigation A coplete navigation syste like those described in the introduction works at different level fro the path planning to reach specific goals. Our ai is only to partially reproduce the portion of the syste required to build a local ap. To this purpose the robot oves on a fixed path consisting of connected third order splines so that the coplete path is continuous and sooth. Figure 4(right) shows an exaple of path represented by a red line. The path following algorith used in this paper is the classical chainedfor controller for unicycle robots (Morin and Sason, 8). The controller require an estiate of the position and orientation errors x e, y e and θ e expressed in the robot reference frae. Given the reference values of linear and angular velocities, respectively u r, and u r,, the control equations are given by { u = u,r K u,r x e u = u,r K u,r y e K u,r tanθ e (6) In our experients we use the following values of control paraeters K =, K =.5 and K =.5. 6 RESULTS The proposed ethod for building sei-static aps has been assessed using both siulation data and experients in the real environent. The tests have been perfored in two siulated environents (Si and Si) and in the hallways of building (Pal) and building (Pal) of the Coputer Engineering Departent of the University of Para. In environent Pal two experients have been executed that are labeled respectively as Pala and Palb.

7 Building (Pal) MARKERS Warning Area Protective Area Mappa Observation Robot Markers (a) 8 9 (b) Building Test B (Palb).5 MARKERS Warning Area Protective Area Map Observation Robot Markers (c) (d) Figure 4: Pictures of experiental environents in Building (Pal) (a) and Building (Pal (c) and the respective polygonal aps (b) and (d). correzione x []. Deviazione standard x Correzione x correzione y [] iterazioni Deviazione standard y Correzione y iterazioni correzione θ [rad] In the latter experients, a person oved in front of the robot to assess how sall objects affect the ap. The results shown in the following are not significantly different fro those achieved in other environents, even without a specific filter for dynaic objects. Siulations have been perfored using the MobileSi siulator provided within the MobileRobots ARIA library that introduces noise both to odoetry and range finder easureents, although not controlled by the user. Real experients have been executed on a MobileRobots Pioneer DX equipped with a Sick LMS laser scanner. Figures 4(a)-(c) illustrate the real environents Pal and Pal. In each environent, the robot repeatedly oved on a fixed path consisting of the union of third order splines. The length of the path is approxiatively 9 in each test run. Obstacles have been placed so that, while the robot oves along its trajectory, such objects partially or totally lie inside the warning or the protective areas. The easureents collected during the first transit of the robot are used to build the local ap of each environent that is used as reference ap for the successive acquisitions. During each next transit, the robot extracts polylines fro each laser scan and associates the to the corresponding polylines of the reference ap according to the proposed algorith. The new polylines are not erged into the reference ap since the purpose of the experient is the assessent of the proposed representation. Figures 4(b)-(d) illustrate the polygonal aps obtained in the experiental tests Pal and Palb..5 Deviazione standard θ Correzione θ iterazioni Figure 5: Corrections perfored by the EKF localizer on state variables x, y and θ during an experient. Environent Si Si Pal Pala Palb Avg Distance [c] Mean Std Dev Min Max Avg Min Max Table : Average, iniu and axiu restricted Hausdorff distances of the polylines w.r.t. reference aps within the respective standard deviation for each siulated and real environent. In the siulation tests, the robot pose has been estiated only by integrating odoetry inforation

8 due to the liitations of the siulator. In particular, the siulated groundtruth is not directly accessible by the application and the laser data do not include the bea reission values aking ipossible to detect reflective arkers. Thus, the EKF localizer described in section 5 can be used only for tests in real environent. Figure 5 illustrates the trend of the correction perfored by the EKF filter on robot state variables x, y and θ for a single test run in Pal within the corresponding σ-bound coputed fro the covariance atrix. The value of correction is bound by or coparable to the σ-bound. Siilar easures have been perfored in all the runs and for all the environents. In all cases, the standard deviation is less than 6 for position and. rad for angular variables. The precision of the polygonal aps is easured by the restricted Hausdorff distances between the corresponding polylines of the reference ap and of current observation. Table reports the results for the different environents. For each polyline i of the reference ap, the ean restricted Hausdorff distance µ i and the corresponding standard deviations σ i have been estiated. The results in the table are respectively the average, iniu and axiu values of ean distances µ i and standard deviations σ i for all the polylines. The ean distance and variance in the siulated environents are greater than in the real environents. This outcoe can be explained with the high accuracy achieved by the localizer. The overall error is always less than c. 7 CONCLUSIONS In this paper, we have presented a ethod to build polygonal local aps that anage sei-static objects in order to iprove the navigation of industrial UGVs. The aps consists of polylines extracted fro laser scans representing the boundary of objects. The proposed ethods extracts, associates and erges the polylines obtained fro the scan into a consistent ap. The proposed data association algorith is based on both the shape siilarity easure and the restricted Hausdorff distance, a novel etric proposed in this work. The accurate environent reconstruction allows the identification of sei-static objects and the definition of efficient navigation policies. To replicate the navigation syste of industrial UGVs, an EKF localizer and a fixed path navigation syste have been ipleented. Experiental results show that the error of the reconstruction is saller than c. In our future works, we expect to exploit the polygonal into a coplete localization and apping algorith. REFERENCES Arras, K. O. (). Feature-Based Robot Navigation in Known and Unknown Environents. PhD thesis, Swiss Federal Institute of Technology Lausanne (EPFL), Thèse No Douglas, D. and Peuker, T. (97). Algoriths for the reduction of the nuber of points required to represent a digitized line or its caricature. Cartographica: Int. Jour. for Geographic Inforation and Geovisualization, ():. 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